## On the two smallest Pisot numbers.(English. Russian original)Zbl 1284.11106

Math. Notes 94, No. 5, 820-823 (2013); translation from Mat. Zametki 94, No. 5, 784-787 (2013).
Summary: Recall that a Pisot number is an algebraic integer greater than $$1$$, whose conjugates lie strictly inside the unit disc $$\mathbb{D} = \{ z \in \mathbb{C}: |z| < 1 \}$$. Pisot numbers can be characterized by their diophantine properties. Consider
$L(\theta): \sup_{\xi \in \mathbb{R}}\liminf_{n \to \infty} \| \xi \theta^n \|$ . The author proves the following theorem: let $$\theta$$ be the greatest root of the equation $$x^3-x-1=0$$, then the equality $$L(\theta)= 1/5$$ holds. Similarly, she proves that if $$\theta$$ is the greatest root of the equation $$x^4-x^3-1=0$$, then the equality $$L(\theta)=3/17$$ holds.

### MSC:

 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11J25 Diophantine inequalities
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### References:

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